where e6da6c268a9789e4eb4e64b8" title="Click to view the MathML source">Δpz:=div(|∇z|p−2∇z), 1<p<n, λ is a positive parameter, 8ed7b76aad25e0a2c735f8841b469a4" title="Click to view the MathML source">r0>0 and 9d56a30bcd8320f8" title="Click to view the MathML source">ΩE:={x∈Rn | |x|>r0}. Here the weight function K∈C1[r0,∞) satisfies 8e741ece054603ce048d523b" title="Click to view the MathML source">K(r)>0 for r≥r0, a3f873dce73529f" title="Click to view the MathML source">limr→∞K(r)=0, and the reaction term ac0b7dd83dfae450" title="Click to view the MathML source">f∈C[0,∞)∩C1(0,∞) is strictly increasing and satisfies e6a3f3497ab947229" title="Click to view the MathML source">f(0)<0 (semipositone), , lims→∞f(s)=∞, and is nonincreasing on [a,∞) for some a>0 and q∈(0,p−1). For a class of such steady state equations it turns out that every nonnegative radial solution is strictly positive in the exterior of a ball, and exists for λ≫1. We establish the uniqueness of this positive radial solution for λ≫1.